Description
Stokes equations for incompressible fluid flow.
This example demonstrates fields defined on subdomains as well as use of periodic boundary conditions.
Find
, such that:
r"""
Stokes equations for incompressible fluid flow.
This example demonstrates fields defined on subdomains as well as use of
periodic boundary conditions.
Find :math:`\ul{u}`, :math:`p` such that:
.. math::
\int_{Y_1 \cup Y_2} \nu\ \nabla \ul{v} : \nabla \ul{u}
- \int_{Y_1 \cup Y_2} p\ \nabla \cdot \ul{v}
= 0
\;, \quad \forall \ul{v} \;,
\int_{Y_1 \cup Y_2} q\ \nabla \cdot \ul{u}
= 0
\;, \quad \forall q \;.
"""
from sfepy import data_dir
from sfepy.fem.periodic import match_y_line
filename_mesh = data_dir + '/meshes/2d/special/channels_symm944t.mesh'
if filename_mesh.find( 'symm' ):
region_1 = {
'name' : 'Y1',
'select' : """elements of group 3""",
}
region_2 = {
'name' : 'Y2',
'select' : """elements of group 4 +e elements of group 6
+e elements of group 8""",
}
region_4 = {
'name' : 'Y1Y2',
'select' : """r.Y1 +e r.Y2""",
}
region_5 = {
'name' : 'Walls',
'select' : """r.EBCGamma1 +n r.EBCGamma2""",
}
region_310 = {
'name' : 'EBCGamma1',
'select' : """(elements of group 1 *n elements of group 3)
+n
(elements of group 2 *n elements of group 3)
""",
}
region_320 = {
'name' : 'EBCGamma2',
'select' : """(elements of group 5 *n elements of group 4)
+n
(elements of group 1 *n elements of group 4)
+n
(elements of group 7 *n elements of group 6)
+n
(elements of group 2 *n elements of group 6)
+n
(elements of group 9 *n elements of group 8)
+n
(elements of group 2 *n elements of group 8)
""",
}
w2 = 0.499
# Sides.
region_20 = {
'name' : 'Left',
'select' : 'nodes in (x < %.3f)' % -w2,
}
region_21 = {
'name' : 'Right',
'select' : 'nodes in (x > %.3f)' % w2,
}
region_22 = {
'name' : 'Bottom',
'select' : 'nodes in (y < %.3f)' % -w2,
}
region_23 = {
'name' : 'Top',
'select' : 'nodes in (y > %.3f)' % w2,
}
field_1 = {
'name' : '2_velocity',
'dtype' : 'real',
'shape' : (2,),
'region' : 'Y1Y2',
'approx_order' : 2,
}
field_2 = {
'name' : 'pressure',
'dtype' : 'real',
'shape' : (1,),
'region' : 'Y1Y2',
'approx_order' : 1,
}
variable_1 = {
'name' : 'u',
'kind' : 'unknown field',
'field' : '2_velocity',
'order' : 0,
}
variable_2 = {
'name' : 'v',
'kind' : 'test field',
'field' : '2_velocity',
'dual' : 'u',
}
variable_3 = {
'name' : 'p',
'kind' : 'unknown field',
'field' : 'pressure',
'order' : 1,
}
variable_4 = {
'name' : 'q',
'kind' : 'test field',
'field' : 'pressure',
'dual' : 'p',
}
integral_1 = {
'name' : 'i1',
'kind' : 'v',
'order' : 2,
}
equations = {
'balance' :
"""dw_div_grad.i1.Y1Y2( fluid.viscosity, v, u )
- dw_stokes.i1.Y1Y2( v, p ) = 0""",
'incompressibility' :
"""dw_stokes.i1.Y1Y2( u, q ) = 0""",
}
material_1 = {
'name' : 'fluid',
'values' : {
'viscosity' : 1.0,
'density' : 1e0,
},
}
ebc_1 = {
'name' : 'walls',
'region' : 'Walls',
'dofs' : {'u.all' : 0.0},
}
ebc_2 = {
'name' : 'top_velocity',
'region' : 'Top',
'dofs' : {'u.1' : -1.0, 'u.0' : 0.0},
}
ebc_10 = {
'name' : 'bottom_pressure',
'region' : 'Bottom',
'dofs' : {'p.0' : 0.0},
}
epbc_1 = {
'name' : 'u_rl',
'region' : ['Left', 'Right'],
'dofs' : {'u.all' : 'u.all', 'p.0' : 'p.0'},
'match' : 'match_y_line',
}
functions = {
'match_y_line' : (match_y_line,),
}
solver_0 = {
'name' : 'ls',
'kind' : 'ls.scipy_direct',
}
solver_1 = {
'name' : 'newton',
'kind' : 'nls.newton',
'i_max' : 2,
'eps_a' : 1e-8,
'eps_r' : 1e-2,
'macheps' : 1e-16,
'lin_red' : 1e-2, # Linear system error < (eps_a * lin_red).
'ls_red' : 0.1,
'ls_red_warp' : 0.001,
'ls_on' : 1.1,
'ls_min' : 1e-5,
'check' : 0,
'delta' : 1e-6,
'is_plot' : False,
'problem' : 'nonlinear', # 'nonlinear' or 'linear' (ignore i_max)
}
save_format = 'hdf5' # 'hdf5' or 'vtk'